Final answer:
The coordinates of the image of a point (h,k) under a half turn about the point A(2,0) are (4-h, -k). This is calculated by reflecting the point across A in both the x and y directions. To reflect a point in the x-axis, use the same x-coordinate and take the opposite of the y-coordinate. To reflect a point in the y-axis, use the same y-coordinate and take the opposite of the x-coordinate.
Step-by-step explanation:
To find the coordinates of the image of a point (h,k) under a half turn about a point A(2,0), we must recognize that a half turn (180 degrees) is a rotation around point A. A half turn will essentially reflect the original point across point A in both the x and y directions.
Firstly, consider the change in the x-coordinate. The distance from the point (h,k) to A(2,0) in the x-direction is h - 2. After a half turn, the image of the point will be the same distance from A but on the opposite side. Therefore, the new x-coordinate will be 2 - (h - 2) or 4 - h.
Similarly, the y-coordinate will undergo the same reflection process. Since A is on the x-axis, the change in the y-coordinate is k - 0 or k. Thus, after the half turn, the new y-coordinate will be 0 - k or -k.
Combining these two coordinate changes, the coordinates of the image of the point after a half turn about A(2,0) are (4 - h, -k).
The coordinate plane is divided into four sections, called quadrants. Quadrant I has positive x and y values, Quadrant II has negative x and positive y, Quadrant III has negative x and y, and Quadrant IV has positive x and negative y.